"Run with it" from "Take the ball and run with it" means to make a decision about something and then do it thoroughly or decisively. I think it's originally a football reference: the player has to take possession of the ball and then run it decisively to the goal, not stand around wondering what to do.
Without more context, I don't know what your sentence means exactly. For example, the speaker might be taking advantage of looser marriage laws in a certain place or religion by marrying someone he/she would not be able to in a different situation (a family member, a person under a certain age, more than one spouse, etc). That is just one guess -- it could be something entirely different. But "run with it" always means to go ahead with a decision without having doubts.
I agree that the origin of the phrase is probably in American football, but "run with it" doesn't apply to most situations where a runner or receiver runs with a ball.
"It's not perfect but it's all you've got-- run with it." When you use this expression, there's a sense of having limited choices, or of better options having run out.
So I think the football analogy refers to a quite specific situation. A quarterback receives the ball at the start of a play, and if he doesn't hand it off immediately, he stands (or maneuvers around) in the backfield looking for a receiver to pass it to. Some members of the other team are trying their best to get to him, and hopefully are blocked from doing so.
If the pass receivers are covered by defenders (who will try to intercept the ball if it's thrown), the quarterback may realize he's out of options and will be tackled very shortly if he doesn't skedaddle-- with the ball, of course. So he runs it into the end zone himself, and scores. .